Subgroups — #28383313

Order 14272.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20297278	C₂	1	1	yes	maximal, characteristic
2	C₂²⋊(C₂₂₃⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
3	#95×C₂₂₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₈₉₂	C₄	1	1	yes	characteristic, complemented
5	C₂×C₂²⋊D₂₂₃	—	2	1	no	—
6	C₂×C₂₂₃⋊C₈	—	2	1	no	—
7	C₄⋊Dic₂₂₃	C₄	1	1	yes	characteristic, complemented
8	(C₂×D₂₂₃)⋊C₄	—	2	1	no	—
9	C₂²×C₄×C₂₂₃	—	2	1	no	—
10	C₂²×C₄×C₂₂₃	C₂²	1	1	yes	characteristic
11	C₂²×D₂₂₃	—	2	1	no	—
12	D₈₉₂	—	4	1	no	—
13	C₂²⋊D₂₂₃	—	4	1	no	—
14	C₂²⋊D₂₂₃	—	4	1	no	—
15	C₂×Dic₂₂₃	—	2	1	no	—
16	C₂₂₃⋊C₈	—	4	1	no	—
17	C₂³×C₂₂₃	D₄	1	1	yes	characteristic
18	C₂×C₄×C₂₂₃	—	2	2	no	—
19	C₂×C₄×C₂₂₃	—	4	1	no	—
20	C₂×C₄×C₂₂₃	—	4	1	no	—
21	C₂×C₄×C₂₂₃	—	4	1	no	—
22	C₂×C₄×C₂₂₃	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×C₂₂₃	D₄	1	1	yes	characteristic
24	C₂²×C₂₂₃	—	2	1	no	—
25	C₂²×C₂₂₃	—	2	1	no	—
26	C₂²×C₂₂₃	C₂²⋊C₄	1	1	yes	characteristic
27	C₂²×C₂₂₃	D₈	1	1	yes	characteristic
28	C₂²×C₂₂₃	SD₁₆	1	1	yes	characteristic
29	C₂×D₂₂₃	—	4	1	no	—
30	C₂×D₂₂₃	—	4	1	no	—
31	C₂×D₂₂₃	—	4	1	no	—
32	C₄×C₂₂₃	—	2	1	no	—
33	C₄×C₂₂₃	—	2	1	no	—
34	C₄×C₂₂₃	—	4	2	no	—
35	Dic₂₂₃	—	4	1	no	—
36	C₂×C₂₂₃	—	2	1	no	—
37	C₂×C₂₂₃	—	2	1	no	—
38	C₂×C₂₂₃	#99	1	1	yes	characteristic
39	C₂×C₂₂₃	#102	1	1	yes	characteristic
40	C₂×C₂₂₃	#104	1	1	yes	characteristic
41	D₂₂₃	—	8	1	no	—
42	C₂₂₃	#327	1	1	yes	characteristic, complemented, Sylow
43	#327	—	223	1	no	maximal, Sylow
44	#121	—	223	1	no	—
45	C₂²⋊C₈	—	223	1	no	—
46	#95	D₂₂₃	1	1	yes	characteristic, complemented
47	C₂×D₄	—	446	1	no	—
48	C₂×D₄	—	223	1	no	—
49	C₂²×C₄	—	2	1	no	—
50	C₂²×C₄	C₂×D₂₂₃	1	1	yes	characteristic
51	C₂×C₈	—	446	1	no	—
52	C₄⋊C₄	—	223	1	no	—
53	C₂²⋊C₄	—	446	1	no	—
54	C₂³	—	446	1	no	—
55	C₂³	C₂²⋊D₂₂₃	1	1	yes	characteristic
56	D₄	—	892	1	no	—
57	D₄	—	892	1	no	—
58	D₄	—	892	1	no	—
59	C₂×C₄	—	2	2	no	—
60	C₂×C₄	—	4	1	no	—
61	C₂×C₄	—	4	1	no	—
62	C₂×C₄	—	4	1	no	—
63	C₂×C₄	—	446	1	no	—
64	C₂×C₄	C₄×D₂₂₃	1	1	yes	characteristic
65	C₂×C₄	D₈₉₂	1	1	yes	characteristic
66	C₈	—	892	1	no	—
67	C₂²	—	2	1	no	—
68	C₂²	—	2	1	no	—
69	C₂²	—	892	1	no	—
70	C₂²	—	892	1	no	—
71	C₂²	—	892	1	no	—
72	C₂²	(C₂×D₂₂₃)⋊C₄	1	1	yes	characteristic
73	C₂²	#16813155	1	1	yes	characteristic
74	C₂²	#16813157	1	1	yes	characteristic
75	C₄	—	2	1	no	—
76	C₄	—	2	1	no	—
77	C₄	—	4	2	no	—
78	C₄	—	892	1	no	—
79	C₂	—	2	1	no	—
80	C₂	—	2	1	no	—
81	C₂	—	1784	1	no	—
82	C₂	#20297218	1	1	yes	characteristic
83	C₂	#20297224	1	1	yes	characteristic
84	C₂	#20297232	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	(C₂×D₂₂₃)⋊C₄
commutator	C₂×C₄×C₂₂₃	C₂×C₄
Frattini	C₂²×C₄	C₂×D₂₂₃
Fitting	#95×C₂₂₃	C₂
socle	C₂²×C₂₂₃	C₂²⋊C₄
prime radical	#20297278	C₂
cosocle	C₂²×C₄×C₂₂₃	C₂²
hypercenter	#95	D₂₂₃
hypocenter	C₂₂₃	#327
LCS	G ▷ C₂×C₄×C₂₂₃ ▷ C₂²×C₂₂₃ ▷ C₂₂₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #95

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₂²⋊(C₂₂₃⋊C₈)	C₂	semidirect
3. #95×C₂₂₃	C₂	semidirect
4. C₂×D₈₉₂	C₄	semidirect
7. C₄⋊Dic₂₂₃	C₄	semidirect
42. C₂₂₃	#327	semidirect
46. #95	D₂₂₃	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#20297278	2
C₂²⋊(C₂₂₃⋊C₈)	2
#95×C₂₂₃	2
#327	223